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A327744 Expansion of Product_{i>=1, j>=1} 1 / (1 - x^(i*j*(j + 1)/2)). 3

%I

%S 1,1,2,4,6,9,17,23,35,54,77,108,163,221,309,436,593,800,1109,1470,

%T 1968,2642,3482,4566,6052,7848,10204,13276,17092,21924,28245,35949,

%U 45762,58231,73609,92789,117140,146799,183826,229995,286483,356040,442566,547489

%N Expansion of Product_{i>=1, j>=1} 1 / (1 - x^(i*j*(j + 1)/2)).

%C Euler transform of A007862.

%H Vaclav Kotesovec, <a href="/A327744/b327744.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: Product_{k>=1} 1 / (1 - x^k)^A007862(k).

%t nmax = 43; CoefficientList[Series[Product[1/(1 - x^k)^Length[Select[Divisors[k], IntegerQ[Sqrt[8 # + 1]] &]], {k, 1, nmax}], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Length[Select[Divisors[d], IntegerQ[Sqrt[8 # + 1]] &]], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 43}]

%t nmax = 50; CoefficientList[Series[Product[1/QPochhammer[x^(k*(k + 1)/2)], {k, 1, Sqrt[2*nmax]}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 24 2019 *)

%Y Cf. A004101, A007862, A327745.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Sep 23 2019

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Last modified October 22 18:45 EDT 2021. Contains 348175 sequences. (Running on oeis4.)